Subgroups — C₂×(C₂×D₃)⋊#672 #14312812

Order 1920.

544 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×D₃×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×#672	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₃)⋊#672	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₂²⋊#672	—	3	1	no	maximal, Hall
6	C₂²×C₄×C₅×D₃	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₅×C₃⋊C₈	C₄	1	1	yes	characteristic
9	C₂×C₅×D₃×C₈	—	2	2	no	—
10	C₂×C₅×D₃×C₈	—	2	2	no	—
11	C₂×C₅×D₃×C₈	C₄	1	2	yes	—
12	C₂×C₅×D₃×C₈	C₂²	1	2	yes	—
13	C₂×C₃⋊#672	—	2	2	no	—
14	C₂×C₃⋊#672	C₂²	1	2	yes	complemented
15	C₂×C₃×#672	—	2	2	no	—
16	C₂×C₃×#672	C₂²	1	2	yes	complemented
17	C₂×(C₂×D₃)⋊C₁₆	—	5	1	no	maximal, Hall
18	C₂³×C₅×C₈	—	3	1	no	—
19	C₂²×#672	—	3	1	no	—
20	C₂²×#672	D₃	1	1	yes	characteristic, complemented
21	C₂²⋊#672	—	3	4	no	—
22	C₂³×C₅×D₃	C₈	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅×Dic₃	C₈	1	1	yes	characteristic
25	C₂×C₄×C₅×D₃	—	2	2	no	—
26	C₂×C₄×C₅×D₃	—	2	2	no	—
27	C₂×C₄×C₅×D₃	C₈	1	2	yes	—
28	C₂×C₄×C₅×D₃	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
31	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
32	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₂×C₅×C₃⋊C₈	—	2	2	no	—
34	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
35	C₅×D₃×C₈	—	2	4	no	—
36	C₅×D₃×C₈	—	2	2	no	—
37	C₅×D₃×C₈	—	2	2	no	—
38	C₃⋊#672	—	2	2	no	—
39	C₃×#672	—	2	2	no	—
40	C₂²×D₃×C₈	D₅	1	1	yes	characteristic
41	C₂²×C₃×C₁₆	—	5	1	no	—
42	C₂²×C₃⋊C₁₆	—	5	1	no	—
43	(C₂×D₃)⋊C₁₆	—	5	4	no	—
44	C₂³×C₄×C₅	—	3	1	no	—
45	C₂²×C₅×C₈	—	3	1	no	—
46	C₂²×C₅×C₈	—	3	2	no	—
47	C₂²×C₅×C₈	—	3	2	no	—
48	C₂²×C₅×C₈	—	6	2	no	—
49	C₂²×C₅×C₈	—	6	2	no	—
50	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
51	C₂×#672	—	2	2	no	—
52	C₂×#672	—	3	2	no	—
53	C₂×#672	—	6	2	no	—
54	C₂×#672	C₂×D₃	1	2	yes	complemented
55	C₂×C₂²⋊C₁₆	—	15	1	no	Sylow
56	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₅×D₃	—	2	2	no	—
58	C₂²×C₅×D₃	—	2	2	no	—
59	C₂²×C₅×D₃	C₁₆	1	4	yes	complemented
60	C₂²×C₅×D₃	C₂×C₈	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
62	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
64	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₅×Dic₃	—	2	2	no	—
66	C₂×C₅×Dic₃	C₂×C₈	1	2	yes	—
67	C₄×C₅×D₃	—	2	4	no	—
68	C₄×C₅×D₃	—	2	2	no	—
69	C₄×C₅×D₃	—	2	2	no	—
70	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
71	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
72	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
73	C₅×C₃⋊C₈	—	2	2	no	—
74	C₂²×C₄×D₃	Dic₅	1	1	yes	characteristic
75	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
76	C₂²×C₃⋊C₈	Dic₅	1	1	yes	characteristic
77	C₂×D₃×C₈	—	2	2	no	—
78	C₂×D₃×C₈	—	2	2	no	—
79	C₂×D₃×C₈	Dic₅	1	2	yes	—
80	C₂×D₃×C₈	C₂×D₅	1	2	yes	—
81	C₂×C₃×C₁₆	—	5	2	no	—
82	C₂×C₃×C₁₆	—	10	2	no	—
83	C₂×C₃⋊C₁₆	—	5	2	no	—
84	C₂×C₃⋊C₁₆	—	10	2	no	—
85	C₂⁴×C₅	—	3	1	no	—
86	C₂²×C₄×C₅	—	3	1	no	—
87	C₂²×C₄×C₅	—	3	2	no	—
88	C₂²×C₄×C₅	—	3	2	no	—
89	C₂²×C₄×C₅	—	6	2	no	—
90	C₂²×C₄×C₅	—	6	2	no	—
91	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
92	C₂×C₅×C₈	—	3	2	no	—
93	C₂×C₅×C₈	—	6	2	no	—
94	C₂×C₅×C₈	—	6	4	no	—
95	C₂×C₅×C₈	—	6	2	no	—
96	C₂×C₅×C₈	—	6	2	no	—
97	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
98	C₂×C₅×C₈	D₁₂	1	2	yes	—
99	C₂×C₅×C₈	#67	1	2	yes	—
100	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
101	#672	—	2	2	no	—
102	#672	—	6	2	no	—
103	C₂³×C₈	—	3	1	no	—
104	C₂²×C₁₆	—	5	1	no	—
105	C₂²×C₁₆	—	15	1	no	—
106	C₂²⋊C₁₆	—	15	4	no	—
107	C₂²×C₃×C₅	C₂²⋊C₈	1	2	yes	—
108	C₂²×C₃×C₅	C₂×C₁₆	1	2	yes	—
109	C₂²×C₃×C₅	M₃₂	1	2	yes	—
110	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₅×D₃	—	2	8	no	—
112	C₂×C₅×D₃	—	2	2	no	—
113	C₂×C₅×D₃	—	2	2	no	—
114	C₂×C₅×D₃	C₂×C₁₆	1	4	yes	complemented
115	C₃×C₄×C₅	C₂²⋊C₈	1	2	yes	—
116	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
118	C₅×Dic₃	—	2	2	no	—
119	C₂³×D₃	#168	1	1	yes	characteristic
120	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
121	C₂²×Dic₃	#168	1	1	yes	characteristic
122	C₂×C₄×D₃	—	2	2	no	—
123	C₂×C₄×D₃	—	2	2	no	—
124	C₂×C₄×D₃	#168	1	2	yes	—
125	C₂×C₄×D₃	C₂×Dic₅	1	2	yes	—
126	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
127	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
128	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
129	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
130	C₂×C₃⋊C₈	—	2	2	no	—
131	C₂×C₃⋊C₈	C₂×Dic₅	1	2	yes	—
132	D₃×C₈	—	2	4	no	—
133	D₃×C₈	—	2	2	no	—
134	D₃×C₈	—	2	2	no	—
135	C₃×C₁₆	—	10	2	no	—
136	C₃⋊C₁₆	—	10	2	no	—
137	C₂³×C₅	—	3	4	no	—
138	C₂³×C₅	—	3	2	no	—
139	C₂³×C₅	—	6	2	no	—
140	C₂³×C₅	—	6	2	no	—
141	C₂³×C₅	D₃×C₈	1	1	yes	characteristic
142	C₂×C₄×C₅	—	3	2	no	—
143	C₂×C₄×C₅	—	6	2	no	—
144	C₂×C₄×C₅	—	6	4	no	—
145	C₂×C₄×C₅	—	6	2	no	—
146	C₂×C₄×C₅	—	6	2	no	—
147	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
148	C₂×C₄×C₅	ΓL(1,𝔽₂₅)	1	2	yes	—
149	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
150	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
151	C₅×C₈	—	6	2	no	—
152	C₅×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
153	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
154	C₅×C₈	C₂×#67	1	1	yes	characteristic
155	C₂³×C₄	—	3	1	no	—
156	C₂²×C₈	—	3	1	no	—
157	C₂²×C₈	—	3	2	no	—
158	C₂²×C₈	—	3	2	no	—
159	C₂²×C₈	—	6	2	no	—
160	C₂²×C₈	—	6	2	no	—
161	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
162	C₂×C₁₆	—	5	2	no	—
163	C₂×C₁₆	—	10	2	no	—
164	C₂×C₁₆	—	15	2	no	—
165	C₂×C₁₆	—	30	2	no	—
166	C₂×C₃×C₅	C₂²⋊C₁₆	1	4	yes	complemented
167	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
168	C₂×C₃×C₅	C₂²×C₁₆	1	1	yes	characteristic
169	C₂×C₃×C₅	C₂×M₃₂	1	1	yes	characteristic
170	C₅×D₃	—	2	4	no	—
171	C₂³×C₃	C₂×#168	1	1	yes	characteristic
172	C₂²×D₃	—	2	2	no	—
173	C₂²×D₃	—	2	2	no	—
174	C₂²×D₃	#672	1	4	yes	complemented
175	C₂²×D₃	C₂×#168	1	2	yes	—
176	C₂×C₃×C₄	C₂×#168	1	1	yes	characteristic
177	C₂×C₃×C₄	#681	1	2	yes	—
178	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
179	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
180	C₂×Dic₃	—	2	2	no	—
181	C₂×Dic₃	C₂×#168	1	2	yes	—
182	C₄×D₃	—	2	2	no	—
183	C₄×D₃	—	2	4	no	—
184	C₄×D₃	—	2	2	no	—
185	C₃×C₈	C₂²⋊Dic₅	1	2	yes	—
186	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
187	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
188	C₃⋊C₈	—	2	2	no	—
189	C₂²×C₅	—	3	4	no	—
190	C₂²×C₅	—	6	2	no	—
191	C₂²×C₅	—	6	8	no	—
192	C₂²×C₅	—	6	2	no	—
193	C₂²×C₅	D₃×C₁₆	1	2	yes	—
194	C₂²×C₅	#798	1	2	yes	—
195	C₂²×C₅	(C₂×D₃)⋊C₈	1	2	yes	—
196	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
197	C₄×C₅	—	6	2	no	—
198	C₄×C₅	(C₂×D₃)⋊C₈	1	2	yes	—
199	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
200	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
201	C₂⁴	—	3	1	no	—
202	C₂²×C₄	—	3	1	no	—
203	C₂²×C₄	—	3	2	no	—
204	C₂²×C₄	—	3	2	no	—
205	C₂²×C₄	—	6	2	no	—
206	C₂²×C₄	—	6	2	no	—
207	C₂²×C₄	D₃×Dic₅	1	1	yes	characteristic
208	C₂×C₈	—	3	2	no	—
209	C₂×C₈	—	6	2	no	—
210	C₂×C₈	—	6	4	no	—
211	C₂×C₈	—	6	2	no	—
212	C₂×C₈	—	6	2	no	—
213	C₂×C₈	D₃×Dic₅	1	1	yes	characteristic
214	C₂×C₈	#1201	1	2	yes	—
215	C₂×C₈	#1203	1	2	yes	—
216	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
217	C₁₆	—	10	2	no	—
218	C₁₆	—	30	2	no	—
219	C₃×C₅	C₂×C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Hall
220	C₂²×C₃	C₂×#672	1	2	yes	—
221	C₂²×C₃	#3944	1	2	yes	—
222	C₂²×C₃	C₂²⋊#168	1	2	yes	—
223	C₂²×C₃	C₂²×#168	1	1	yes	characteristic
224	C₂×D₃	—	2	8	no	—
225	C₂×D₃	—	2	2	no	—
226	C₂×D₃	—	2	2	no	—
227	C₂×D₃	C₂×#672	1	4	yes	complemented
228	C₃×C₄	C₂²⋊#168	1	2	yes	—
229	C₃×C₄	C₂×#681	1	1	yes	characteristic
230	C₃×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
231	Dic₃	—	2	2	no	—
232	C₂×C₅	—	6	4	no	—
233	C₂×C₅	(C₂×D₃)⋊C₁₆	1	4	yes	complemented
234	C₂×C₅	C₂×D₃×C₁₆	1	1	yes	characteristic
235	C₂×C₅	C₂×#798	1	1	yes	characteristic
236	C₂×C₅	C₂×(C₂×D₃)⋊C₈	1	1	yes	characteristic
237	C₂³	—	3	4	no	—
238	C₂³	—	3	2	no	—
239	C₂³	—	6	2	no	—
240	C₂³	—	6	2	no	—
241	C₂³	D₃×#168	1	1	yes	characteristic
242	C₂×C₄	—	3	2	no	—
243	C₂×C₄	—	6	2	no	—
244	C₂×C₄	—	6	4	no	—
245	C₂×C₄	—	6	2	no	—
246	C₂×C₄	—	6	2	no	—
247	C₂×C₄	D₃×#168	1	1	yes	characteristic
248	C₂×C₄	#6650	1	2	yes	—
249	C₂×C₄	(C₂×D₃)⋊Dic₅	1	2	yes	—
250	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
251	C₈	—	6	2	no	—
252	C₈	(C₂×D₃)⋊Dic₅	1	2	yes	—
253	C₈	C₂×#1201	1	1	yes	characteristic
254	C₈	C₂×#1203	1	1	yes	characteristic
255	C₂×C₃	C₂²⋊#672	1	4	yes	complemented
256	C₂×C₃	C₂²×#672	1	1	yes	characteristic
257	C₂×C₃	C₂×#3944	1	1	yes	characteristic
258	C₂×C₃	C₂×C₂²⋊#168	1	1	yes	characteristic
259	D₃	—	2	4	no	—
260	C₅	C₂×(C₂×D₃)⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
261	C₂²	—	3	4	no	—
262	C₂²	—	6	8	no	—
263	C₂²	—	6	2	no	—
264	C₂²	—	6	2	no	—
265	C₂²	D₃×#672	1	2	yes	—
266	C₂²	#90935	1	2	yes	—
267	C₂²	(C₂×D₃)⋊#168	1	2	yes	—
268	C₂²	C₂×D₃×#168	1	1	yes	characteristic
269	C₄	—	6	2	no	—
270	C₄	(C₂×D₃)⋊#168	1	2	yes	—
271	C₄	C₂×#6650	1	1	yes	characteristic
272	C₄	C₂×(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
273	C₃	C₂×C₂²⋊#672	1	1	yes	characteristic, complemented, Sylow
274	C₂	—	6	4	no	—
275	C₂	(C₂×D₃)⋊#672	1	4	yes	direct product factor
276	C₂	C₂×D₃×#672	1	1	yes	characteristic
277	C₂	C₂×#90935	1	1	yes	characteristic
278	C₂	C₂×(C₂×D₃)⋊#168	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₈	D₃×D₅
commutator	C₂×C₃×C₅	C₂²×C₁₆
Frattini	C₂×C₈	C₂×D₃×D₅
Fitting	C₂²×C₃×C₅×C₈	C₂²
socle	C₂³×C₃×C₅	C₂×C₈
prime radical	C₂³×C₅×D₃	C₈
cosocle	C₂×C₃×C₅×C₈	C₂³
hypocenter	C₃×C₅	C₂×C₂²⋊C₁₆
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂²×C₃⋊#672	C₂	semidirect
3. C₂²×C₃×#672	C₂	semidirect
4. (C₂×D₃)⋊#672	C₂	direct (N × H)
14. C₂×C₃⋊#672	C₂²	semidirect
16. C₂×C₃×#672	C₂²	semidirect
20. C₂²×#672	D₃	semidirect
54. C₂×#672	C₂×D₃	semidirect
59. C₂²×C₅×D₃	C₁₆	semidirect
114. C₂×C₅×D₃	C₂×C₁₆	semidirect
166. C₂×C₃×C₅	C₂²⋊C₁₆	semidirect
174. C₂²×D₃	#672	semidirect
219. C₃×C₅	C₂×C₂²⋊C₁₆	semidirect
227. C₂×D₃	C₂×#672	semidirect
233. C₂×C₅	(C₂×D₃)⋊C₁₆	semidirect
255. C₂×C₃	C₂²⋊#672	semidirect
260. C₅	C₂×(C₂×D₃)⋊C₁₆	semidirect
273. C₃	C₂×C₂²⋊#672	semidirect
275. C₂	(C₂×D₃)⋊#672	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₅×D₃×C₈	2
C₂²×C₃⋊#672	2
C₂²×C₃×#672	2
(C₂×D₃)⋊#672	2
C₂×C₂²⋊#672	3
C₂×(C₂×D₃)⋊C₁₆	5

Subgroups — C2×(C2×D3)⋊#672 #14312812

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×(C₂×D₃)⋊#672 #14312812